On continuity of measurable group representations and homomorphisms
Let $G$ be a locally compact group, and let $U$ be its unitary representation on a Hilbert space $H$. Endow the space $\mathcal L(H)$ of bounded linear operators on $H$ with the weak operator topology. We prove that if $U$ is a measurable map from $G$ to $\mathcal L(H)$ then it is continuous. This result was known before for separable $H$. We also prove that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous.